Open string instantons and superpotentials

We study the F terms in N = 1 supersymmetric, d = 4 gauge theories arising from D(p + 3)-branes wrapping supersymmetric p-cycles in a Calabi-Yau threefold. if p is even the spectrum and superpotential for a single brane are determined by purely classical (alpha' --> 0) considerations. If p = 3, superpotentials for massless modes are forbidden to all orders in alpha' and may only be generated by open string instantons. For this latter case we find that such instanton effects are generically present. Mirror symmetry relates even and odd p and thus perturbative and nonperturbative superpotentials; we provide a preliminary discussion of a class of examples of such mirror pairs.